| Gantry crane is a class of common transport equipments which is widely used to convey heavy or hazardous loads in a variety of industrial environments.By applying anti-swing control strategies to gantry crane systems,the unmanned,automatic and intelligent degree of transportation operations is improved such that the operation efficiency is improved and the labor cost is reduced.Due to the existence of flexible cable,the mathematical model of the considered gantry crane needs to be described by partial differential equations(PDEs).However,the infinite-dimensional characteristic in PDE-based control systems makes it difficult in theoretic analysis.Therefore,research on anti-swing control of gantry cranes based on PDE models not only holds significant theoretical importance but also has extensive practical applications.In this study,a comprehensive and systematic investigation is conducted on a gantry crane governed by PDE.Four anti-swing control approaches are proposed as follows:backstepping-based state-feedback boundary control,backstepping-based output-feedback boundary control,boundary control for the crane with external disturbance,and eventtriggered boundary control.The well-posedness and stability of the closed-loop systems under the four methods are analyzed.The main contributions of this study can be summarized as follows:(1)The state-feedback anti-swing boundary control for a class of gantry cranes with flexible cable is studied.Based on the backstepping approach,a state-feedback boundary controller is designed to push the trolley such that the trolley drives the load to the desired position with less shaking through flexible steel cables.The considered control strategy transforms the complex original system into a simple target system through the designed kernel functions and backstepping transformations.The well-posedness of the kernel functions is proven,which ensures the one-to-one correspondence between the states of the original system and the target system.On this basis,the well-posedness and exponential stability of the original system are obtained based on these of the target system.(2)An output-feedback control strategy is proposed for the considered gantry crane via backstepping-based approach.This method,based on only boundary measurements,can estimate all of the spatial states rapidly.Compared with the state-feedback boundary controller,this approach can reduce hardware costs and make the controller more applicable in practical applications.Based on the operator semigroup theory and the Lyapunov method,the target system is proven to be well-posed and stable.Then,using the backstepping technique,the closed-loop system is also proven to be well-posed and stable.(3)For the gantry crane with external disturbance,an anti-swing boundary controller is designed,which can resist the influence of disturbance effectively.Firstly,by using the Hamilton’s principle and calculus of variations,the PDE model of the gantry crane with external disturbance is constructed.Next,an auxiliary function is developed to realize the estimate of external disturbance.Then,based on the estimate of external disturbance,an anti-swing boundary controller is proposed,which only uses boundary measurements and can effectively suppress the influence of disturbance and the vibration during the transport process.Finally,the solutions of the closed-loop system are proven to be well-posedness and uniformly ultimately bounded.(4)An event-triggered anti-swing boundary control approach is developed.Compared with continuous control methods,this approach significantly reduces the update number of the controller,reduces the communication frequency between the controller and the actuator,and ensures superior control performance at the same time,by only boundary measurements.For the closed-loop system under the proposed event-triggered boundary control,it is strictly proven that the existence of a positive inter-event time is established,the Zeno behavior can be avoided and there exists a unique solution to the system.Finally,using the Lyapunov method,the states of the gantry crane system are proven to be ultimately bounded. |
